A104625 Expansion of 1/(sqrt(1-4*x) - x^2).

1, 2, 7, 24, 87, 322, 1211, 4604, 17645, 68042, 263655, 1025632, 4002601, 15662422, 61427543, 241386924, 950160607, 3745589510, 14784496003, 58424093536, 231112008371, 915065382154, 3626113490579, 14379912928572, 57064644495359

Offset: 0

Author

Paul Barry, Mar 17 2005

Keywords

• easy
• nonn

Comments

Diagonal sums of convolution triangle of central binomial coefficients A054335.

Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (2,2). - Alois P. Heinz, Sep 14 2016

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000984, A026671, A054335.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(sqrt(1-4*x) - x^2))); // G. C. Greubel, Aug 12 2018

Mathematica

CoefficientList[Series[1/(Sqrt[1-4*x] -x^2), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)

PARI

x='x+O('x^50); Vec(1/(sqrt(1-4*x) - x^2)) \\ G. C. Greubel, Aug 12 2018

Formula

Conjecture: n*a(n) + (n-3)*a(n-1) + 2*(-28*n+51)*a(n-2) + 72*(2*n-5)*a(n-3) - n*a(n-4) + (-5*n+3)*a(n-5) + 18*(2*n-5)*a(n-6) = 0. - R. J. Mathar, Feb 20 2015

a(n) = Sum_{k=0..floor(n+2)/2} 4^(n+2-2*k) * binomial(n+1-3*k/2,n+2-2*k). - Seiichi Manyama, Feb 06 2024